Support Operator Method Derivation: Outline

The Support Operator Method for Diffusion on Hexahedra:

• Represent the diffusion term ( $\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$}$D$\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$$\Phi$ ) as the
  divergence ( $\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$} \cdot$}$ ) of a gradient ( $\mbox{$\mbox{$\stackrel{^{\mathstrut}\smash{\longrightarrow}}{\nabla}$}$}$ )
• Explicitly define one of the operators (in this case, the divergence operator)
• Define the remaining operator (in this case, the gradient operator) as the discrete adjoint of the first operator
• The previous step is accomplished by discretizing a portion of a vector identity

In other words, the first operator is set up explicitly, and the second operator is defined in terms of the first operator's definition.